# Apples to Apples

What I'm about to write might sound like blasphemy to those of you who have been reading my column for years.  But, here goes - it is NOT all about expected value.  There I said it.  But an explanation has to go with it.  No, I'm not talking about the importance of liking a game you are playing, which can also override the payback to a certain extent.  In this case, I am not talking about any emotional issues, but a purely mathematical one.  Let me explain.

For years, my father pounded into everyone's heads that when you play video poker, you play the hand with the highest expected value.  For the past 10+ years, I have done the same.  Nothing we've said was wrong.  It is just that in video poker, once you start a hand, your wager is constant.  Thus, the expected value is expressed as a percent of your wager that you will have returned to you, on average, in the long run.  If you were dealt a Low Pair, the expected value is 0.82, meaning you can expect to have 82% of your original wager returned.  If you were playing 1 nickel, you can expect to get back just over 4 cents in the long run.  If you were playing max-coin \$1, you can expect about \$4.10 on average in the long run.  So, if you are dealt a 4-Card Straight at the same time (with 1 High Cards) with an expected value of 0.74, we are comparing apples and apples.  You'll get back 82% of your wager with the Low Pair and 74% with the 4-Card Straight.  Since the wager is the same in both cases, it is fairly obvious which one you would want back.

While most situations in a casino involve identical sized wagers, not all do.  The most common situation occurs in Blackjack.  When you double down, you double the size of your wager in the middle of the hand.  This is not the same as making a Play wager in something like Three Card Poker.  If Three Card Poker allowed you to 'check' and simply have the showdown with the Dealer as opposed to folding, then it would be a bit more similar, but still not the same.  Blackjack's double down creates a relatively unique situation.  Normally, if we are more likely to win that to lose we want to bet everything we can.  So, in a game like 4-Card Poker, when you are a favorite to win the hand, you want to play 3X.  But, you're not giving up anything for the privilege to do so.  In Blackjack, when you decide to double down, you are limiting yourself to one and only card for a hit.

In many cases, this is all you would want to do.  If you have an 11 against a Dealer 6, you'd never take more than 1 card anyhow.  So, in this case you're not giving up anything.  But what about doubling into a 9 with an 11.  You get dealt a 2.  Now you're stuck with 13.  If you hadn't doubled down, you'd be hitting another card.  In Four Card Poker, whether you Play 3x or choose to Play only 1x, the probability of you winning with your particular hand doesn't change.  In blackjack, the decision to double down CAN change your probability of winning.  It won't against a 6, but it most definitely will against a 9.

A simple simulation proves my point.  Whether I double down or hit against a 6, The Player will have an expected value of 1.34.  So he can either win 134% of \$5 or 134% of \$10.  But if we simply say the expected value of both situation is 1.34 then it sounds as if the decision is a tossup when it is not.  If we look at the situation against the 9, the situation gets more confusing.  The expected value of hitting is 1.16.  The expected value of doubling is only 1.11.  So, does this mean we should hit and not double?  ABSOLUTELY NOT.  The wagers are not equal so we really need to look at the net wagers won or lost over time.   If we face this situation 1 million times (okay, probably not in several lifetimes), if we hit, we win a net of 157,740 wagers.  If we double, we win 228,022 wagers.  In this case, a 'wager' is whatever our initial bet was.  By doubling, we increase our net win by about 50%.  We don't get the full 100% that we do against a 6, because our inability to hit again means we give back some of what we gain by being able to double.  But, there is no doubt that we want to double even though it has a lower expected value.

I've covered a similar topic over the years when discussing different paytables for different denominations in video poker.  A quarter machine may have a payback 0.5% higher than a nickel machine.  But, because you are betting 5 times as much, you'll find that you'll actually lose more playing the quarter machine.  Payback is merely the overall expected value of a game.  As such, similar principles apply to a game's payback as does to a particular play's expected value.  You need to make sure you are comparing apples to apples.  Most of the time, in the casino, you are.  But there are exceptions.   Next week, I'll discuss how this concept is applied to Ultimate Texas Hold'em which has one of the most unique betting patterns of any game in the casino.